We will show that the total right derived functor \(\mathbb {R}S_e \colon \mathrm {h}{H}_\infty ^{i I} \longrightarrow \mathrm {h}{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  is an equivalence of categories.
If \(R^{i I} \colon {H}_\infty ^{i I} \rightarrow {H}_\infty ^{i I}\)  is a fibrant replacement functor in \({H}_\infty ^{i I}\) , we can write \(\mathbb {R}S_e\)  as the composition [1]}
\(\begin{tikzcd}\mathrm {h}{H}_\infty ^{i I} [r, "\mathrm {h}R^{i I}"]& \mathrm {h}({H}_\infty ^{i I})_f [r, "\mathrm {h}S_e"]& {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\,.\end{tikzcd}\) 
